Place squash on a baking sheet; drizzle with olive oil. Season with salt and pepper.

Bake in the preheated oven until tender, about 45 minutes. Remove and cool. Cut into cubes.

Heat butter in a skillet over medium heat; cook and stir shallot and garlic in the melted butter until shallot is transparent, 5 to 10 minutes. Add sun-dried tomatoes; cook until softened, 2 to 3 minutes. Mash squash cubes into shallot mixture until relatively smooth. Remove from heat, transfer mixture to a bowl, and cool for 2 to 3 minutes.

Stir bread crumbs, egg, and Parmesan cheese into squash mixture. Add more bread crumbs if mixture is too sticky. Season with salt and pepper. Shape mixture into 4 patties.

Heat vegetable oil in a large frying pan over medium-high heat; cook patties in the hot oil until browned, 4 to 5 minutes per side.